Integral inequalities for the growth and higher derivative of polynomials

Inequalities for the polar derivative and the generalized polar derivative of complex polynomials with restricted zeros

“…There is always a desire to look for better and improved bounds than those available in this domain. It is this aspiration of obtaining more refined and revamped bounds that has inspired our work in this article .In this paper, we have generalized and refined some well known results concerning the polynomials due to Tur𝑎ǹ [21], Rather and Dar [19] , Rather, Ali ,Shafi and Dar [18] and others. (1.1)…”

“…If 𝑃(𝑧) is a polynomial of degree 𝑛 having all its zeros in |𝑧| ≤ 𝑘 , 𝑘 ≥ 1 , then for any complex 𝛼 with 1 + 𝑘 + 𝑘 𝑛 The above Lemma due Rather el at. [18].…”

“…Recently, Rather el at. [18] consider the lower bound estimates for the generalized polar derivative of certain polynomials, which include various results due to Aziz and Rather , Turaǹ and Govil as special cases.…”

“…Recently, Rather el at. [18] extended inequality (1.9) to the generalized polar derivative of a polynomial by showing that, if 𝑃 ∈ Ƥ 𝑛 has all its zeros in |𝑧| ≤ 𝑘 , 𝑘 ≥ 1 , then for every 𝛼 ∈…”